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Let K be an open set of $\mathbb{R}^n$, then I know that I can get a countable cover by using closed cuboids $Q_i$ with a pairwise disjoint interior.

I want to show that by this fact every open set is measurable under the Lebegues measure. I suppose that I need to show that every cuboid is measurable (under the Lebesgue measure). But I don't know exactly how to proceed. Do you have a clue?

Rico1990
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    In order to show that a cuboid $Q$ is Lebesgue measurable you have to check Carathéodory's criterion ( https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_criterion). One inequality is trivial (from sub-additivity) and for the other one you must use the very definition of (outer) Lebesgue measure. If you need it, I can give you more hints. – Onil90 Jul 18 '18 at 13:28
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    @Onil90 ... thank you for providing a hint and not a solution. But I wonder how long our over-eager solvers can resist? – GEdgar Jul 18 '18 at 14:26
  • Yes, thank you. I got it. – Rico1990 Jul 19 '18 at 10:14

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